Can I bake the animation of a mesh and get rid of its armature?
The word for that kind of thing is "vertex animation". Blender supports shape keys that can be imported into unity via FBX. They are typically used for cloth animation and facial animations that are difficult to do with bones - if you're looking for movement in general then you'd generally want to stick to the armature and use an armature in Unity as well. You could also try to use a different armature in Unity and retarget the animations with Mechanim - if the armatures are compatible Mechanim usually works great.

If you just converted the keyframed poses to shape keys you might get an animation that looks roughly like the armature-based animation, but shape keys don't interpolate the same way that bones do. If you have to use shape keys you might have to create a shape key for every frame and change the blend shape in Unity in every frame.

You can create a shape key by selecting "Apply as Shape Key" in the armature modifier of the mesh that's controlled by the armature:


This thread might be of intererest to you:
https://blenderartists.org/forum/showthread.php?300572-Is-it-possible-to-bake-armature-poses-into-shape-keys-morph-targets